Adding a skirt / border to object with sharp corners leaves gaps
Import the svg curve.

Scale it to some size that is workable.


SVGs are imported as curve objects. You don't need to convert them to mesh in order to extrude, you can do that in the geometry section>extrude.



(I changed the color for visibility to the blue state that Texas should be)

Then create a new curve object, I will name it "Lip Curve" that has the profile shape you are looking for. Make it small, much smaller than the other object. To make sharp angles select two control points, press V to access the handle type menu, and select "vector"





Use the "Lip Curve" as taper object for the Texas Outline.



You can clean up the object a bit by lowering the resolution for the curve (in this case I lowered the preview resolution for the curve from 16 to 1)



Note that I'm getting a double lip on the object because this particular curve was created thicker in the vector drawing program (inkscape, illustrator, etc) So when it is imported into blender that thickness will be made out of two parallel curves to give it thickness. 



To further refine the shape and do cleanup of overlapping segments convert to mesh and manually fix imperfections.